Question

find dy/dx, using implicit differentiation
6xy-y/5=2/x

Answer 1

\[\frac{d}{dx}(6xy-\frac{y}{5})=\frac{d}{dx}\frac{2}{x}\]
\[6(y+x \frac{dy}{dx})-\frac{1}{5}\frac{dy}{dx}=-\frac{2}{x^2}\]
Solve for dy/dx
\[\frac{dy}{dx}(6x-\frac{1}{5})=-\frac{2}{x^2}-6y \implies \frac{dy}{dx}(6x-\frac{1}{5})=\frac{-2-6x^2y}{x^2} \implies \frac{dy}{dx}=\frac{-2(1+3x^2y)}{x^2(6x-\frac{1}{5})}\]
\[\frac{dy}{dx}=\frac{-10(1+3x^2y)}{x^2(30x-1)}\]
Normally you insert what y is:
\[6xy-y/5=2/x \implies y=\frac{10}{30x-1}\]
I'll let you do the algebra :P

Answer 2

thank you

Answer 3

No problem